STA256H5S LEC0101-LEC0102, Summer 2022 Assignment 3
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STA256H5S LEC0101-LEC0102, Summer 2022
Assignment 3
Q1 Let f (x, y) = 2 for 0 < x < y, 0 < y < 1, and 0 elsewhere, be the joint pdf of (X, Y). Show that the correlation coefficient of X and Y is ρ = .
Q2 Let X1 , X2 , and X3 be random variables with equal variances but with correlation coefficients ρ 12 = 0.5, ρ 13 = 0.2, and ρ23 = 0.3. Find the correlation coefficient of the linear functions Y = X1 + X2 and Z = X2 + X3 .
Q3 Consider a shipment of 1000 items into a factory. Suppose the factory can tolerate about 5% defective items. Let X be the number of defective items in a sample without replacement of size n = 10. Suppose the factory returns the shipment if X > 3.
(a) Obtain the probability that the factory returns a shipment of items that has 5%
defective items.
(b) Suppose the shipment has 8% defective items. Obtain the probability that the
factory returns such a shipment.
(c) Obtain approximations to the probabilities in parts (a) and (b) using appro- priate binomial distributions.
Q4 On the average, a grocer sells three of a certain article per week. How many of these should he have in stock so that the chance of his running out within a week is less than 0.05? Assume a Poisson distribution.
Q5 Suppose the lifetime in months of an engine, working under hazardous conditions, has a Gamma distribution with a mean of 5 months and a variance of 10 months squared.
(a) Determine the median lifetime of an engine.
(b) Suppose such an engine is termed successful if its lifetime exceeds 10 months. In
a sample of 10 engines, determine the probability of at least 2 successful engines.
Q6 Let X1 and X2 be independent random variables with normal distributions N (6, 1) and N (7, 1), respectively. Find P (X1 > X2 ).
Hint: Write P (X1 > X2 ) = P (X1 - X2 > 0) and determine the distribution of X1 - X2 .
Q7 Let the random variable Yn have a distribution that is b(n, p). Prove that 1 -
converges in probability to 1 - p.
2022-08-11